Beyond expected utility: Subjective risk aversion and optimal portfolio choice under convex shortfall risk measures

Abstract

We conduct a decision-theoretic analysis of convex shortfall risk measures regarding their flexibility to represent subjective risk aversion, and discuss the implications for the choice of optimal portfolios. As convex shortfall risk measures are closely related to the expected utility functional, we draw upon the expected utility framework as our benchmark. First, we show that, unlike expected utility, convex shortfall risk measures do always represent constant absolute risk aversion. This constitutes a significant limitation when changing initial wealth of, e.g., investors, is relevant. Interestingly, though, it is exactly this limitation that provides additional degrees of freedom in representing subjective risk aversion beyond expected utility when initial wealth is fixed, as, e.g., in returned-based portfolio selection models in the tradition of Markowitz (1952). Second, we apply convex shortfall risk measures to the standard portfolio problem between a riskless and a risky asset. We provide a procedure that allows inferring the optimal portfolios under convex shortfall risk measures from certain optimal expected utility-portfolios. The procedure incorporates the additional degrees of freedom in modeling subjective risk aversion and, thus, allows more flexible and realistic portfolios compared to expected utility. Third, we address the optimal portfolio choice between a riskless and a risky asset in the presence of an additional background risk. Again, we prove a correspondence result between optimal portfolios under convex shortfall risk measures and expected utility and obtain more flexible and realistic portfolios compared to expected utility. We also compare the optimal risky investments without and with background risk and provide a necessary and sufficient condition for a reduction of the risky investment.